Compensator for compensation of higher-order chromatic dispersion

ABSTRACT

A chromatic-dispersion compensator comprises a plurality of dispersive elements (Q in number. Q≧2) each exhibiting a dispersion characteristic D (λ) that varies substantially as a polynominal function of wavelength, the polynominal function being of an order (P−2) greater than 0, the dispersion characteristics being displaced in wavelength relative to each other such that the compensator has a net Pth-order dispersion characteristic D P  that is non-zero and does not vary substantially with wavelength over an operating bandwidth.

[0001] This invention relates to the field of compensators forcompensation of higher-order chromatic-dispersion and in particular tosuch compensators for use in optical-fibre networks. “Higher-orderchromatic dispersion” refers to chromatic dispersion of an order Phigher than second order (i.e. P>2).

[0002] Wave-division-multiplexed (WDM) networks are importantcommunications systems. As channel bit-rates have increased, the problemof temporal dispersion in dense WDM networks has become an increasinglyimportant consideration in system design. There is a need for devicesproviding dynamically variable, uniform dispersion compensation over alarge bandwidth.

[0003] Dynamic second-order dispersion compensation has beendemonstrated using, for example, fibre Bragg gratings (FBGs) [B. J.Eggleton et al., IEEE Photonics Tech. Lett. 11(7), 854 (1999)], atunable etalon [L.D. Garrett, Proc. OFC 2000, Paper PD7, Baltimore, Md.,March 2000], and a device based on a Gires-Tournois interferometer [C.K. Madsen, G. Lenz, Proc. OFC 2000, Paper WF5, Baltimore, Md., March2000]. Such devices can be essentially viewed as periodic time-sampledsystems. The inventors have already proposed the use of concatenatedAWGs for compensation of second-order chromatic dispersion (see, forexample, M. C. Parker, S. D. Walker, “Virtually ripple-free,multi-channel, adaptive dispersion compensator based on are-multiplexing AWG cascade”, Proc. ECOC2000, Paper P2.7, Munich,September 2000).

[0004] An object of the present invention is to provide a compensatorfor compensation of higher-order (P>2) chromatic dispersion thatprovides uniform higher-order dispersion compensation over a largebandwidth. The standard formula for the Pth-order filter dispersioncharacteristic D_(P) with filter phase characteristic θ(ω), where ω isthe angular frequency, is given by: $\begin{matrix}{D_{P} = {{\frac{\partial^{P - 1}\quad}{\partial\lambda^{P - 1}}\left\lbrack \frac{\partial{\theta (\omega)}}{\partial\omega} \right\rbrack} \approx {\frac{\lambda_{0}^{2}}{2\pi \quad c}\frac{\partial^{P}{\theta (\lambda)}}{\partial\lambda^{P}}}}} & (1)\end{matrix}$

[0005] where c is the speed of light and λ₀ is the wavelength at thepass-band centre.

[0006] At the pass-band centre wavelength λ₀, the overall dispersion Dmay be approximated by a Taylor expansion about λ₀:

D(λ)=D′ ₂ +D′ ₃(λ−λ₀)+D′₄(λ−λ₀)² + . . . +D′ _(P)(λ−λ₀)^(P−2)

[0007] Here D′_(P)≈D_(P) at the centre wavelength λ₀.

[0008] According to the invention there is provided achromatic-dispersion compensator comprising a plurality of dispersiveelements (Q in number, Q≧2) each exhibiting a dispersion characteristicD(λ) that varies substantially as a polynomial function of wavelength,the polynomial function being of an order (P−2) greater than 0 (i.e.(P−2)>0), the dispersion characteristics being displaced in wavelengthrelative to each other such that the compensator has a net Pth-orderdispersion characteristic D_(P) that is non-zero and does not varysubstantially with wavelength over an operating bandwidth.

[0009] The dispersion characteristic D(λ) may vary substantially as apolynomial because, for example, it varies as a polynomial exhibiting aripple, such as a sinusoidal ripple. Thus the dispersion characteristicD(λ) may vary as a polynomial when any ripple is disregarded.

[0010] It is well known that second order dispersion can be compensatedfor by a parabolic phase profile (i.e. linear chirp), but compensationof higher-order dispersions is a non-trivial problem. In the context ofsecond order dispersion, a quadratic phase function${\phi (m)} = {\gamma \left( {m - \frac{M}{2}} \right)}^{2}$

[0011] is mathematically represented as a complex Gaussian function$\exp {\left\lfloor {{- j}\quad {\gamma \left( {m - \frac{M}{2}} \right)}^{2}} \right\rfloor.}$

[0012] When Fourier transformed, a Gaussian function conveniently yieldsanother Gaussian function, so that the filter phase characteristic dueto a linearly-chirped grating is also purely quadratic (parabolic) atthe centre of the filter response. Taking the appropriate seconddifferential of the filter parabolic phase response characteristic givesa constant value and hence a uniform second order dispersioncharacteristic.

[0013] However, the Fourier transform (FT) of a higher-order phaseprofile Φ(m)=γ|m−m/2|^(P) (complex exponential function) does not simplyyield another complex exponential function with the appropriately higherorder phase profile, in contrast to the special case of a parabolicphase (Gaussian function) which Fourier transforms to another Gaussianfunction with a purely parabolic phase profile. Instead, solutions tothe FT of higher-order phase functions involve confluent hypergeometricand Whittaker functions. Those functions tend to have polynomial phasefunctions with all lower orders of dispersion present as well. Hence, asingle grating device with a higher-order phase profile (such as thatproposed by Lowery—A. J. Lowery, P. C. R. Gurney, Proc. OFC '99, PaperFD5, San Diego, Calif., February 1999) will tend to be dispersive forall lower orders as well; i.e. it will not dispersion compensate solelyat the dispersion order of interest. It will also exhibit high ripple inthe dispersion characteristic. To achieve higher-order dispersioncompensation for a particular dispersion order only (or to achievedispersion compensation in a controlled manner for all the lower ordersas well), it is necessary to concatenate devices (appropriately detunedwith respect to one another) to balance out the lower orders ofdispersion (or achieve the desired level of dispersion at eachparticular order), and reduce the ripple in the dispersion order ofinterest.

[0014] Thus the higher-order dispersion characteristics of theindividual dispersive elements may be shifted relative to each other insuch a way that, for example, any ripple on the Pth-order dispersioncharacteristic D_(P) is cancelled out by propagation through all of theelements of the compensator. For example, if there are four identicaldispersive elements having such ripple, their Pth-order dispersioncharacteristics D_(P) will be shifted by one quarter of a periodrelative to each other, so that troughs and peaks in the ripples ofelements cancel out overall.

[0015] Of course, it is sufficient for the characteristic to enable thenet Pth-order dispersion D_(P) of the compensator to not varysubstantially with wavelength over the operating bandwidth; variationsoutside that bandwidth are generally irrelevant.

[0016] The Pth-order dispersion characteristics D_(P) of each of theelements may be of substantially the same form; however, they may bescaled in magnitude and/or wavelength. The number Q of dispersiveelements required and the wavelength shifts required will depend uponthose scalings; for example, it may be possible to replace an element inany particular compensator with two elements each having Pth-orderdispersion characteristics D_(P) of half the magnitude of the originalelement. Preferably, each element's Pth-order dispersion characteristicD_(P) varies periodically with a period K. In general, the flat responsecan be built up by an appropriate choice of magnitudes, periods anddisplacements for the dispersive elements in much the same way as afunction can be built up by an appropriate choice of sine and cosinewaves in Fourier analysis. Thus, concatenation of dispersive elementswith non-identical dispersion characteristics and wavelength shifts thatare non-integer multiples of K/Q (including zero) is possible to effectsuitable flattening of Pth-order dispersion in the operating bandwidth.

[0017] Preferably, the compensator has a net Rth-order dispersion thatis non-zero for all R<P (for R>2 and P>3). Alternatively, thecompensator may have a net Rth-order dispersion D_(R) of zero for atleast one R<P (for R>2 and P>3). The compensator may have a netRth-order dispersion of zero for all R<P (for R>2 and P>3). Preferably,half of the dispersive elements exhibit Rth-order dispersion DRcharacteristics that are the negative of the Rth-order dispersioncharacteristics D_(R) of the other half of the dispersive elements.

[0018] Each dispersive element may exhibit a dispersion characteristicof substantially the same form as that of each other dispersive element.Each dispersive element may exhibit a dispersion characteristic ofsubstantially the same magnitude as that of each other dispersiveelement. Alternatively, the dispersion characteristics may be of thesame form but of a different magnitude.

[0019] Preferably, the chromatic-dispersion compensator comprises meansfor actively varying the dispersion with time. Preferably, each of thedispersive elements comprises the means for actively varying thedispersion with time. A substantially polynomial variation in opticalpath length of polynomial order P will produce a grating chirpsubstantially of polynomial order (P−1).

[0020] Preferably, the chromatic-dispersion compensator comprises meansfor actively varying the wavelength displacement with time. Preferably,the centre wavelength of the compensator can be tuned by a linearvariation with wavelength in the optical path length through thecompensator. The optical path lengths may be varied by, for example,thermal means, electrical means or mechanical means.

[0021] Preferably, the plurality of dispersive elements form a pluralityof groups, each group comprising a plurality of dispersive elements, andin which the dispersion characteristics of the dispersive elementswithin each group are displaced in wavelength relative to each other andthe dispersion characteristics of each group are displaced in wavelengthrelative to each other.

[0022] More preferably, there are two dispersive elements within eachgroup, and there are two groups, and the dispersion characteristics ofthe elements of a first of the groups are displaced in wavelengthrelative to each other by an amount and the dispersion characteristicsof the elements of a second of the groups are also displaced inwavelength relative to each other by that amount.

[0023] The dispersive elements may be polynomially chirped-gratingdevices. The dispersive elements may be Fibre Bragg gratings.Preferably, the Fibre Bragg gratings are interconnected by a circulator.Alternatively, the dispersive elements may be arrayed-waveguide gratings(AWGs). An AWG typically comprises first and second free-propagationregions (which may comprise, for example, silica for a silica-based AWG)and an array of waveguides interconnecting the first and second freepropagation regions, the optical path lengths of any two adjacentwaveguides being different. In general, the optical path lengths ofadjacent channels increase linearly across the waveguide, butalternatively the optical path lengths of adjacent channels may increasenonlinearly across the waveguide. Alternatively, the optical pathlengths of adjacent channels may increase between some adjacent channelsand decrease between other adjacent channels across the waveguide.

[0024] Preferably, adjacent AWGs will have adjacent free-propagationregions that are connected to each other by N waveguides. Morepreferably, the waveguides have entrance and exit apertures lying onarcs. Still more preferably, there are apertures at the boundary betweenadjacent free-propagation regions. An AWG having a single input port canbe considered to be a 1×N de-multiplexer and a second such AWG, adjacentto the first, as an N×1 re-multiplexer. N represents the number of portsat the interface between the adjacent AWGs. In a single AWG, N will bethe number of output ports from the device as a whole; for concatenatedAWGs, however, N becomes a free design parameter, because the ports areinternal to the device, and need not even correspond to actualapertures. Thus, N may be chosen to allow the compensator to be tailoredfor optimum insertion loss and physical size.

[0025] Preferably, an active trapezoidal region on the AWG imparts thewavelength displacement. Preferably, the means for varying thedispersion of the compensator is an active region on the AWG having ageometric shape corresponding to the phase polynomial θ(λ) of order P.

[0026] Preferably, the chromatic-dispersion compensator comprises oneinput waveguide. Preferably, the chromatic-dispersion compensatorcomprises one output waveguide. Preferably, the chromatic-dispersioncompensator comprises many input waveguides. Preferably, thechromatic-dispersion compensator comprises many output waveguides.

[0027] According to another aspect of the invention, there is provided achromatic-dispersion compensator comprising two pairs of two dispersiveelements, each element exhibiting a Pth-order dispersion characteristicD_(P) exhibiting a ripple, one dispersive element within each pairhaving a Rth-order (R<P, R>2, P>3) dispersion characteristic that is thenegative of the dispersion characteristic D_(R) of the other dispersiveelement within that pair, the dispersion characteristics of one pair ofelements being displaced from a centre wavelength by an amountproportional to a first amount (T₂−T₃) and a second amount (T₂+T₃)respectively and the dispersion characteristics of the other pair ofelements being displaced from a centre wavelength by an amountproportional to a third amount (−T₂−T₃) and a fourth amount (−T₂+T₃)respectively, T₂ and T₃ being such that the net Pth-order dispersion ofthe compensator is non-zero and does not vary substantially withwavelength over an operating bandwidth.

[0028] According to another aspect of the invention there is provided amethod of changing the phase of light as a function of frequency,comprising propagating the light through a chromatic-dispersioncompensator described above as according to an aspect of the invention.

[0029] According to another aspect of the invention, there is provided amethod of compensating the dispersion of light arising from propagationin a dispersive optical system, comprising propagating light through achromatic-dispersion compensator described above as according to anaspect of the invention.

[0030] According to another aspect of the invention, there is provided amethod of providing dispersion compensation, the method comprising:passing light of a plurality of wavelengths through a plurality ofdispersive elements (Q in number, Q>2) and dispersing the light in eachdispersive element by an amount that varies substantially as apolynomial function of wavelength D(λ), the polynomial function being ofan order (P−2) greater than 0, the dispersive elements exhibitingdispersion characteristics that are displaced in wavelength relative toeach other such that, in passing through all of the elements, the lightis dispersed by a net Pth-order dispersion D_(P) that is non-zero anddoes not vary substantially with wavelength over an operating bandwidth.

[0031] According to another aspect of the invention, there is provided amethod of dispersion compensation, the method comprising: (i) providingtwo pairs of two dispersive elements each exhibiting a Pth-orderdispersion characteristic exhibiting a ripple, one dispersive elementwithin each pair having an Rth-order (R<P, R>2, P>3) dispersioncharacteristic D_(R) that is the negative of the dispersioncharacteristic of the other dispersive element within that pair, thedispersion characteristics of one pair of elements being displaced froma centre wavelength by an amount proportional to a first amount (T₂−T₃)and a second amount (T2+T₃)respectively and the dispersioncharacteristics of the other pair of elements being displaced from acentre wavelength by an amount proportional to a third amount (−T₂−T₃)and a fourth amount (−T₂+T3) respectively; (ii) setting T2 to zero andoptimising T₃ to reduce Rth-order dispersion D_(R) to zero; (iii)optimising T₂ to overlap the ripples to provide a non-zero Pth-orderdispersion D_(P) that does not vary substantially with wavelength overan operating bandwidth.

[0032] An embodiment of the invention will now be described, by way ofexample only, with reference to the accompanying drawings, of which:

[0033]FIG. 1 is a schematic showing a dispersion compensator accordingto the invention;

[0034]FIG. 2 shows simulated second-order dispersion characteristics asfunctions of wavelength for the compensator of FIG. 1 optimised forthird-order compensation:

[0035] (a) first and second AWG within a pair (solid and dashed linesrespectively)

[0036] (b) first and second AWG pair (solid and dashed linesrespectively)

[0037] (c) full device

[0038]FIG. 3 shows simulated third-order dispersion characteristics asfunctions of wavelength for the compensator of FIG. 1 optimised forthird-order compensation:

[0039] (a) first and second AWG within a pair (solid and dashed linesrespectively)

[0040] (b) first and second AWG pair (solid and dashed linesrespectively)

[0041] (c) full device

[0042]FIG. 4 shows simulated composite characteristics as functions ofwavelength for the compensator of FIGS. 1 optimised for third-ordercompensation: (a) normalised transmission |t(λ)|² (dB), (b) group delayτ_(d) (ps), (c) second-order dispersion characteristic D₂(λ) (ps nm⁻¹)(as FIG. 2) and (d) third-order dispersion characteristic D₃(λ) (psnm⁻²) (as FIG. 3).

[0043]FIG. 5 shows simulated second-order dispersion characteristics asfunctions of wavelength for the compensator optimised for fourth-ordercompensation:

[0044] (a) first and second AWG within a pair (solid and dashed linesrespectively)

[0045] (b) first and second AWG pair (solid and dashed linesrespectively)

[0046] (c) full device

[0047]FIG. 6 shows simulated third-order dispersion characteristics asfunctions of wavelength for the compensator of FIG. 1 optimised forfourth-order dispersion compensation:

[0048] (a) first and second AWG within a pair (solid and dashed linesrespectively)

[0049] (b) first and second AWG pair (solid and dashed linesrespectively)

[0050] (c) full device.

[0051]FIG. 7 shows simulated fourth-order dispersion characteristics asfunctions of wavelength for the compensator of FIG. 1 optimised forfourth-order dispersion compensation:

[0052] (a) first and second AWG within a pair (solid and dashed linesrespectively)

[0053] (b) first and second AWG pair (solid and dashed linesrespectively)

[0054] (c) full device.

[0055]FIG. 8 shows simulated composite characteristics as functions ofwavelength for the compensator of FIG. 1 optimised for fourth-ordercompensation: (a) normalised transmission |t(λ)|² (dB), (b) second-orderdispersion characteristic D₂(λ) (ps nm⁻¹) (as FIG. 5), (c) third-orderdispersion characteristic D₃(λ) (ps nm⁻²) (as FIG. 6) and (d)fourth-order dispersion characteristic D₄(λ) (ps nm⁻³) (as FIG. 7).

[0056]FIG. 9 shows a chirped fibre Bragg grating carousel based on a6-port optical circulator, suitable for third-or fourth-order dispersioncompensation.

[0057] The device shown in FIG. 1 consists of four AWGs 10-40 formingpair A and pair B (pair B is not shown in detail but is substantiallyidentical to pair A). Each AWG comprises: free-propagation regions 50,60 (having arcuate boundaries, but depicted with linear boundaries forease of representation); waveguide array 70; trapezoidal active region80; and polynomial active region 90. The characteristics, such asfree-spectral range (FSR) and number of arrayed waveguides, of AWGs10-40 are identical, except that the AWGs' dispersion-compensationwavelength profiles are slightly detuned with respect to one another.(Of course, it is not a requirement of the invention that allcharacteristics of the dispersive elements be identical, for example,the FSRs may be integer multiples of each other, for example, to achievea non-integer K/Q.)

[0058] Note that polynomial active region 90′ in AWG 20 (and AWG 40) isconvex whereas polynomial active region 90 in AWG 10 (and AWG 30) isconcave. The active regions 90, 90′ are thus “negatives” of each other,which results in the dispersion characteristics of AWG 10 beingcancelled out by those of AWG 20, for at least some dispersive orders.

[0059] Operation of AWG 10 will now be described; in this embodiment,operation of the other AWGs is substantially identical, save for thedetuning.

[0060] An AWG can be considered to be made up of two free propagationregions, one on the input side and one on the output side of the AWG,which are interconnected by an array of M+1 waveguiding channels, insequence m=0 to M, with the channels generally having graduallyincreasing optical path lengths, so that the optical path length of themth channel is greater than that of the (m−1)th channel. Lightcomprising single or multiple wavelengths Σλ_(i) is transmitted alongoptical fibre 100 and then propagates through free-propagation region 50until it reaches the waveguide array 70. Free-propagation regions 50 and60 are sufficiently long to allow Fraunhofer diffraction to occur, whichmeans that Fourier optical concepts can be employed in the analysis ofthe AWG [M. C. Parker et al., IEEE Journal of Special Topics in QuantumElectronics on Fibre-optic Passive Components, 5(5), 1379 (1999)].Waveguide array 70 can be regarded as a Fourier plane within the opticalsystem.

[0061] The input light is distributed with a Gaussian envelope amplitudeprofile$E_{0}{\exp \left\lbrack {- {\alpha \left( {m - \frac{M}{m}} \right)}^{2}} \right\rbrack}$

[0062] across the waveguides of array 70. Polynomial active region 90 isa phase-control means that can be used to produce a programmable phaseprofile in the array 70 (which is the Fourier plane), which results in aquasi-linear, quasi-quadratic, quasi-cubic or higher-order chirp thatexhibits ripple in the device response spectrum.

[0063] Active trapezoidal region 80 is a phase-control means used toimpose across the array a programmable linear phase profile. The Fouriertransform of that imposed profile is a wavelength shift, which manifestsitself at plane 110 after propagation away from the Fourier planethrough free-propagation region 60.

[0064] Each of active regions 80, 90 may for example be a layer ofhydrogenated amorphous silicon (α-Si:H) for an AWG based on silicontechnology or alternatively a thermo-optic region for an AWG based onsilica. Alternatively, the regions may be embodied for example in theform of electrodes in an AWG based on indium phosphide or lithiumniobate technology. The phase shift imparted on a given waveguide willbe proportional to the optical path length of the channel segment overwhich the phase control means extends; hence, a cubic phase shift isimparted by an active region having an optical path length varyingcubically across the array 70 and a linear phase shift is imparted by anactive region having an optical path length varying linearly across thearray 70 (for example, a trapezoidal region).

[0065] AWG 10 can be considered to be a 1×N de-multiplexer and AWG 20 tobe an N×1 re-multiplexer. By designating the free spectral range of theAWGs 10-40 such that FSR=N×200 GHZ, the device shown in FIG. 1 operatesas an in-line variable dispersion compensator on all 200 GHZ-ITU-gridchannels. N represents the number of ports at the interface between AWG10 and AWG 20 and so is a free design parameter (as is the number ofports at the interface between AWG 30 and AWG 40), such that the overalldevice can be tailored for optimum insertion loss and physical size(which tends to scale as approximately 1/FSR).

[0066] Pairs 10, 20 and 30, 40 of polynomially chirped AWGs,appropriately detuned with respect to each other, thus have a positiveand negative polynomial (of order P) phase profile (equivalent to anonlinear chirp of order P-1) imposed, by polynomial active regions 90,90′, on their Fourier planes such that the opposite disposition of thechirps ensures a zero level of lower-order dispersions at the centre ofthe passband. However, anti-symmetric (i.e. differential) detuning ofeach AWG 10-40 via the trapezoidal active region 80 causes thedispersion order of interest to remain finite. Polynomial active regions90 are symmetric about the longitudinal axis of the AWG so that nodetuning occurs. However, it would also be possible to havenon-symmetric polynomial active regions 90, 90′ and to use trapezoidalactive regions 80 to cancel out any resulting detuning.

[0067] For the kth output port of a single AWG, the spectraltransmission response is given approximately by $\begin{matrix}{{t_{k}(\lambda)} \approx {\frac{{- j}\sqrt{\pi \quad}{rW}}{\lambda \quad R}{\sum\limits_{m = 0}^{M}{\exp \left\{ {- {\alpha \left( {m - \frac{M}{2}} \right)}^{2}} \right\} \exp \left\{ {{j{\frac{2\pi \quad n\quad \Delta \quad l}{\lambda}\left\lbrack {1 + {A\left( V_{a} \right)} - \frac{{Wx}_{k}}{R\quad \Delta \quad 1}} \right\rbrack}m} + {j\frac{{2\pi \quad n\quad \Delta \quad l}\quad}{\lambda}{C\left( V_{c} \right)}{{m - \frac{M}{2}}}^{P}}} \right)}}}} & (2)\end{matrix}$

[0068] where n is the refractive index, Δl the incremental path lengthdifference between neighbouring waveguides in an equivalent devicehaving no active regions, r is the waveguide mode spot size, R is thelength of the free propagation region (FPR), W is the centre-to-centredistance between neighbouring waveguides at the FPR entrance, M+1 is thenumber of waveguides in the array of each AWG, X_(k) is the distance ofthe kth output port from the optical axis and a is the apodisationparameter associated with the assumed Gaussian envelope electric-fieldamplitude profile across the arrayed waveguides. The integer value ofthe polynomial order P is the same as the dispersion order to becorrected, i.e. P=2, 3, 4 for second-, third-, fourth-order dispersioncompensation respectively. All C-AWGs (chirped AWGs) are assumed to haveidentical physical and geometric parameters. Analytic solutions to theintegral version of equation (2) only exist under tight constraints andgenerally involve confluent hypergeometric and Whittaker functions. Thevoltage-dependent coefficient A(V_(a)) tunes (in trapezoidal region 80)the centre wavelength of light at the kth output port λ_(0,k) such that$\begin{matrix}{\lambda_{0,k} \approx {\sqrt{{{FSRn}\quad \Delta \quad l}\quad}\left\lbrack {1 + {A\left( V_{a} \right)} - \frac{x_{k}W}{R\quad \Delta \quad l}} \right\rbrack}} & (3)\end{matrix}$

[0069] By considering an AWG as a planar 4f lens-relay system,Fourier-Fresnel transform theory can be employed [M. C. Parker et al.,IEEE Journal of Special Topics in Quantum Electronics on Fibre-opticPassive Components, 5(5), 1379 (1999)].

[0070] The voltage-dependent polynomial coefficient C(V_(c)) acts onlyto control the degree of nonlinear chirping and hence the strength ofdispersion compensation. A symmetric phase region ensures that the C-AWGspectral characteristic is not detuned as C(V_(c)) is varied.

[0071] The product of the individual AWG transfer functions gives theoverall device response. By appropriate spectral displacement of theAWGs relative to each other, the overall Pth-order dispersion is givenby the summation of the individual dispersion characteristics, suchthat: $\begin{matrix}{{D_{P}^{overall}(\lambda)} = {\sum\limits_{i = 1}^{Q}{D_{Pi}\left( {\lambda + {\Delta \quad \lambda_{i}}} \right)}}} & (4)\end{matrix}$

[0072] where Q is the total number of AWGs and Δλ_(i), is the spectraldisplacement for the ith AWG and is an optimisation parameter. Detuningis achieved via parameter A(V_(a)), which is different for each AWG10-40 and equals: AWG # Detuning 10 A = T₁ + T₂ − T₃ 20 A = T₁ + T₂ + T₃30 A = T₁ − T₂ − T₃ 40 A = T₁ − T₂ + T₃

[0073] T₁ corresponds to the centre wavelength of the AWG operatingpass-band. The method of calculating appropriate values of T₂ and T₃ fora substantially flat, non-zero third-order-dispersion characteristic isas follows:

[0074] (i) set T₂ to zero and optimise T₃ to reduce second-orderdispersion to zero;

[0075] (ii) optimise T₂ to overlap the ripples in the third-ordercharacteristic to provide a non-zero third-order dispersion that doesnot vary substantially with wavelength over the desired operatingbandwidth.

[0076] Surprisingly, the inventors have found that the same method canbe used with a device having a quartic phase profile (P=4) to optimisefor a substantially flat, non-zero fourth-order-dispersioncharacteristic. Specifically, the method is to:

[0077] (i) set T₂ to zero and optimise T₃ to reduce third-orderdispersion to zero;

[0078] (ii) optimise T₂ to overlap the ripples in the fourth-ordercharacteristic to provide a non-zero fourth-order dispersion that doesnot vary substantially with wavelength over the desired operatingbandwidth.

[0079] Surprisingly, the second-order dispersion is also reduced to zeroby this method of optimising for fourth-order dispersion.

[0080] We simulated a concatenation of AWGs for adaptive third- andfourth-order compensation at 100 Gb/s. Each AWG has a FSR=153.6 nm(96×200 GHz), with M=128 waveguides in each array. For example, N=48apertures in the interfaces between the AWG re-multiplexing pairs wouldachieve compensation for all channels spaced by 400 GHz.

[0081] FIGS. 2 to 4 show the spectral characteristics of a deviceproviding third-order dispersion compensation and also show howoptimised detuning can produce smooth overall dispersioncharacteristics. FIG. 2(a)-(c) shows how the (a) individual, (b) pairedand (c) overall AWG second-order dispersion varies at the pass-bandcentre. It can be seen from FIG. 2(b) that the linear characteristic,passing through zero dispersion, of FIG. 2(c) results from the fact thatthe characteristic of one AWG pair is the inverse of that of the other(i.e. one transforms into the other by successive reflection about thepass-band centre wavelength and about zero dispersion). FIG. 3 shows thecorresponding third-order dispersion characteristics. The flat overallthird-order response near the pass-band centre (FIG. 3(c)) results fromrelative displacement of the third-order characteristics of pair A andpair B (FIG. 3(b)) by about 1.5 nm. Ripples in the characteristics ofeach pair are thus combined to give a flat response. The displacedcentral ‘M’-shaped portion of the characteristic of the first pair(solid line, FIG. 3(b)) can be seen to result from superposition of theindividual characteristics of AWGs 10 and 20 (FIG. 3(a)).

[0082] The optimised values for third-order compensation (P=3) areT₁=−1.54×10⁻², T₂=3.3×10⁻⁴, T₃=4.4×10⁻⁴. Each AWG has been parabolicallychirped to achieve maximum dispersion compensation, such that$\begin{matrix}{{C\left( V_{c} \right)}_{\max} = {\left( \frac{2}{M} \right)^{P}\frac{FSR}{2\pi \quad \lambda_{0}}F_{M}}} & (5)\end{matrix}$

[0083] where $F_{M} = {\frac{7}{5}\pi}$

[0084] is the phase change accumulated along the longest waveguide inthe array. Here the Gaussian parameter is α=0.8. FIG. 4 indicates thatthe 3-dB width of the passband is 141 GHz. Simulated insertion loss isat most 5.5 dB per AWG. A symmetric, parabolic group delay τ_(d)characteristic (FIG. 4(b)) ensures a linear second-order dispersioncharacteristic through zero (FIG. 4(c)). The third-order dispersioncharacteristic (FIG. 4(d)) provides almost ripple-free compensation ofup to 4.15 ps nm⁻² (±0.05 ps nm⁻²) over a 100 GHz range centred onλ₀=1560 nm. That would compensate multiple 100 Gb/s WDM channels for upto 70 km of single-mode fibre, assuming 0.06 ps nm⁻² km⁻¹ third-orderdispersion.

[0085] FIGS. 5 to 8 show simulation results for a P=4 quartic-phaseprofile (i.e. cubically chirped) AWG cascade, where the AWGs have thesame physical characteristics as before.

[0086]FIG. 5 (a)-(c) shows how the (a) individual and (b) paired AWGssecond-order dispersion characteristics combine to produce zerosecond-order dispersion at the pass-band centre (FIG. 5(c)). In contrastto the locally linear characteristic of FIG. 2(c), the characteristicsin this case is locally quadratic about the pass-band centre, touchingzero dispersion at its local minimum. FIG. 5(b) shows that thecharacteristics of one AWG pair is the mirror image of that of the otherAWG pair, reflected at the centre wavelength of the pass-band.

[0087]FIG. 6(c) shows that the quadratic second-order characteristicresults in a locally linear third-order characteristic passing throughzero at the pass-band centre. Thus zero second- and third- orderdispersion at the pass-band centre corresponds to the second-ordercharacteristic being locally a quadratic, having a zero, and also havinga stationary point, at the pass-band centre. FIG. 6(b) shows that thelinear characteristic, passing through zero of FIG. 6(c) alsocorresponds to the characteristic of one AWG pair being the inverse ofthat of the other.

[0088] The M-shaped curve of the fourth-order dispersion characteristic(FIG. 7(c)) results from the fourth-order characteristic of each pairbeing the mirror image of the other and displaced in wavelength fromeach other. The fourth-order characteristic of Pair A (solid line, FIG.7(b)) can be seen to result from the superposition of thecharacteristics of AWGs 10 and 20 (FIG. 7(a)).

[0089] We see from FIG. 8 that, for fourth-order compensation, the 3-dBwidth of the passband is now reduced to 116 GHz, and the fourth-orderdispersion characteristic (FIG. 8(d)) provides up to 11.3 ps nm⁻³compensation, with ±0.6 ps nm⁻³ ripple in the 3 dB passband. Theoptimised values for fourth-order compensation (P=4) are T₁=−1.54×10⁻²,T₂=1.9×10⁻⁴, T₃=4.3×10⁻⁴.

[0090] The AWG cascade of FIG. 1 is equivalent to a carousel of fibreBragg gratings 260, 270, 280, 290 (operating in a reflective mode)around a 6-port optical circulator 230, as shown in FIG. 9. In general,for a dispersion compensator device consisting of Q chirped FBGs(dispersive elements) a ‘Q+2’-port optical circulator is required, sincetwo extra ports for the input and output waveguides 240, 250 are alsorequired. (It should be noted that higher-port-count optical circulatorsare easily made by appropriately concatenating multiple lower-port-countoptical circulators.) The two groups 210, 220 each consist of two FBGs(260, 270; 280,290), one 260, 280 positively chirped and one 270, 290negatively polynomially chirped. The strength of chirping is given bythe value assigned to F_(M). Equation (5) relates the value of theappropriate FBG chirp F_(M) to the equivalent value of the coefficientC(V_(c))_(max) in a chirped AWG. The FBGs 260-290 are detuned withrespect to each other by appropriate amounts ±δλ±Δλ (equivalent to theAWG detunings proportional to ±T₂±T₃ associated with the AWG embodiment)from the centre wavelength λ₀, to achieve the appropriate flat third- orfourth-order dispersion characteristics over the pass-band range ofinterest. We note that since AWGs tend to operate at high gratingorders, the small free-spectral range (FSR) allows multiple wavelengthsto be dispersion compensated. This would imply that long period FBGsoperating at similarly high-orders would also be suitable for multiplewavelength use.

[0091] In FIGS. 2 to 9, we have only simulated low-ripple compensationusing two-phase concatenation of grating pairs. However, polyphaseconcatenation is also possible to reduce ripple still further.

[0092] The device described above may be employed as a fine-tuningdispersion compensation element in conjunction with a fixed dispersioncompensation device (e.g. dispersion compensation fibre), itselfcompensating for the second- and higher-order dispersion of a fixedlength of 100 km of single-mode fibre. Such a device can be usefullyemployed in long-haul submarine or terrestrial systems, where automaticdispersion correction is a desirable feature. It will be appreciatedthat various modifications and variations can be made to the designsdescribed above.

1. A chromatic-dispersion compensator comprising a plurality of dispersive elements (Q in number, Q≧2) each exhibiting a dispersion characteristic D(λ) that varies substantially as a polynomial function of wavelength, the polynomial function being of an order (P−2) greater than 0, the dispersion characteristics being displaced in wavelength relative to each other such that the compensator has a net Pth-order dispersion characteristic D_(P) that is non-zero and does not vary substantially with wavelength over an operating bandwidth.
 2. A chromatic-dispersion compensator as claimed in claim 1, in which the compensator has a net Rth-order dispersion that is non-zero for all R<P (for R≧2 and P>3).
 3. A chromatic-dispersion compensator as claimed in claim 1, in which the compensator has a net Rth-order dispersion of zero for at least one R<P (for R ≧2 and P≧3).
 4. A chromatic-dispersion compensator as claimed in claim 3, in which the compensator has a net Rth-order dispersion of zero for all R<P (for R≧2 and P≧3).
 5. A chromatic-dispersion compensator as claimed in any preceding claim, in which each dispersive element exhibits a dispersion characteristic of substantially the same form as that of each other dispersive element.
 6. A chromatic-dispersion compensator as claimed in any preceding claim, in which each dispersive element exhibits a dispersion characteristic of substantially the same magnitude as that of each other dispersive element.
 7. A chromatic-dispersion compensator as claimed in any preceding claim, in which half of the dispersive elements exhibit Rth-order (R<P) dispersion characteristics that are the negative of the Rth-order dispersion characteristics of the other half of the dispersive elements.
 8. A chromatic-dispersion compensator as claimed in any preceding claim, in which each element's Pth-order dispersion characteristic varies periodically.
 9. A chromatic-dispersion compensator as claimed in any preceding claim, comprising means for actively varying the dispersion with time.
 10. A chromatic-dispersion compensator as claimed in claim 9, in which each of the dispersive elements comprises the means for actively varying the dispersion with time.
 11. A chromatic-dispersion compensator as claimed in any preceding claim, comprising means for actively varying the wavelength displacement with time.
 12. A chromatic-dispersion compensator as claimed in any preceding claim, in which the centre wavelength of the compensator can be tuned by a linear variation with wavelength in the optical path length through the compensator.
 13. A chromatic-dispersion compensator as claimed in any preceding claim, in which the plurality of dispersive elements form a plurality of groups, each group comprising a plurality of dispersive elements, and in which the dispersion characteristics of the dispersive elements within each group are displaced in wavelength relative to each other and the dispersion characteristics of each group are displaced in wavelength relative to each other.
 14. A chromatic-dispersion compensator as claimed in claim 13, in which there are two dispersive elements within each group, and there are two groups, and in which the dispersion characteristics of the elements of a first of the groups are displaced in wavelength relative to each other by an amount and the dispersion characteristics of the elements of a second of the groups are displaced in wavelength relative to each other by that amount.
 15. A chromatic-dispersion compensator as claimed in any preceding claim, in which the dispersive elements are polynomially chirped-grating devices.
 16. A chromatic-dispersion compensator as claimed in claim 15, in which the dispersive elements are fibre Bragg gratings.
 17. A chromatic-dispersion compensator as claimed in claim 16, in which the fibre Bragg gratings are interconnected by a circulator.
 18. A chromatic-dispersion compensator as claimed in claim 15, in which the dispersive elements are arrayed-waveguide gratings (AWGs).
 19. A chromatic-dispersion compensator as claimed in claim 18, in which adjacent AWGs will have adjacent free-propagation regions that are connected to each other by waveguides.
 20. A chromatic-dispersion compensator as claimed in claim 19, in which the waveguides have entrance and exit apertures lying on arcs.
 21. A chromatic-dispersion compensator as claimed in claim 20, in which there are apertures at the boundary between adjacent free-propagation regions.
 22. A chromatic-dispersion compensator as claimed in any of claims 18 to 21, in which an active trapezoidal region on the AWG imparts the wavelength displacement.
 23. A chromatic-dispersion compensator as claimed in any of claims 18 to 21 as dependent on claim 9, in which the means for varying the dispersion of the compensator is an active region on the AWG having a geometric shape corresponding to the phase polynomial of order P.
 24. A chromatic-dispersion compensator as claimed in any preceding claim, comprising one input waveguide.
 25. A chromatic-dispersion compensator as claimed in any preceding claim, comprising one output waveguide.
 26. A chromatic-dispersion compensator as claimed in any preceding claim, comprising many input waveguides.
 27. A chromatic-dispersion compensator as claimed in any preceding claim, comprising many output waveguides.
 28. A chromatic-dispersion compensator comprising two pairs of two dispersive elements, each element exhibiting a Pth-order dispersion characteristic D_(P) exhibiting a ripple, one dispersive element within each pair having a Rth-order (R<P, R≧2 and P≧3) dispersion characteristic D_(R) that is the negative of the dispersion characteristic of the other dispersive element within that pair, the dispersion characteristics of one pair of elements being displaced from a centre wavelength by an amount proportional to a first amount (T₂−T₃) and a second amount (T₂+T₃) respectively and the dispersion characteristics of the other pair of elements being displaced from a centre wavelength by an amount proportional to a third amount (−T₂−T₃) and a fourth amount (−T₂+T₃) respectively, T₂ and T₃ being such that the net Pth-order dispersion of the compensator is non-zero and does not vary substantially with wavelength over an operating bandwidth
 29. A method of changing the phase of light as a function of frequency, comprising propagating the light through a chromatic-dispersion compensator as claimed in any of claims 1 to
 28. 30. A method of compensating the dispersion of light arising from propagation in a dispersive optical system, comprising propagating light through a chromatic-dispersion compensator as claimed in any of claims 1 to
 28. 31. A method of providing dispersion compensation, the method comprising: passing light of a plurality of wavelengths through a plurality of dispersive elements (Q in number, Q≧2) and dispersing the light in each dispersive element by an amount that varies substantially as a polynomial function of wavelength D(λ), the polynomial function being of an order (P−2) greater than 0, the dispersive elements exhibiting dispersion characteristics that are displaced in wavelength relative to each other such that, in passing through all of the elements, the light is dispersed by a net Pth-order dispersion D_(P) that is non-zero and does not vary substantially with wavelength over an operating bandwidth.
 32. A method of dispersion compensation, the method comprising: (i) providing two pairs of two dispersive elements each exhibiting a Pth-order dispersion characteristic exhibiting a ripple, one dispersive element within each pair having a Rth-order (R<P, R≧2 and P≧3) dispersion characteristic D_(R) that is the negative of the dispersion characteristic of the other dispersive element within that pair, the dispersion characteristics of one pair of elements being displaced from a centre wavelength by an amount proportional to a first amount (T₂−T₃) and a second amount (T₂+T₃) respectively and the dispersion characteristics of the other pair of elements being displaced from a centre wavelength by an amount proportional to a third amount (−T₂−T₃) and a fourth amount (−T₂+T₃) respectively; (ii) setting T₂ to zero and optimising T₃ to reduce Rth-order dispersion D_(R) to zero; (iii) optimising T₂ to overlap the ripples to provide a non-zero Pth-order dispersion D_(P) that does not vary substantially with wavelength over an operating bandwidth.
 33. A dispersion compensator substantially as herein described with reference to the accompanying drawings.
 34. A method of providing dispersion compensation substantially as described herein, with reference to the accompanying drawings. 